The Cyberium was an AI which contained all the knowledge and future history of all Cybermen, (TV: The Haunting of Villa Diodati) serving a role once held by the history computer. (TV: The Tomb of the Cybermen, PROSE: Iceberg, et al.)
The Alliance sent the Cyberium back through time and space in order to prevent the return of the Cyberman Empire. In the 21st century, Captain Jack Harkness informed the Thirteenth Doctor's companions, Graham O'Brien, Yasmin Khan and Ryan Sinclair, about this, but did not specify exactly what the Alliance had sent back, though he did send the Thirteenth Doctor a warning, telling her to beware of the Lone Cyberman and not give it what it wanted. (TV: Fugitive of the Judoon)
While out walking in June 1816, Percy Shelley spotted the Cyberium in Lake Geneva. When he fished it out of the water, it melted in his hand and entered his bloodstream. Upon returning to Villa Diodati, the Cyberium tried to hide itself by making Shelley invisible to everyone else and even utilized a perception filter to keep from being discovered. Meanwhile, Shelley acted as its guardian. At the same time, the Thirteenth Doctor and her companions were visiting Mary Godwin, Percy Shelley, Lord Byron and Claire Clairmont in Villa Diodati, while the Lone Cyberman had invaded the house in search of the Cyberium. Though a long-standing enemy of the Cyberman, this was the Doctor's first encounter with the Cyberium; indeed she initially displayed an unfamiliarity with the term.
When the Doctor was finally able to locate Shelley, she had him tell her what happened. She noticed that the Cyberium was burning through his mind and would destroy him if it stayed in him much longer. She referred to this as "the Cyberium at the heart of it, controlling data, strategy, decision-making" and a "super-computer fused to a cerebral cortex". The Cyberman demanded that the group surrendered the Cyberium, but in order to be extracted, the Cyberium had to leave willingly. The Doctor proceeded to use an old Time Lord trick to show Shelley his future death, which tricked the Cyberium into leaving his body. It floated in mid-air, before flowing into the Doctor. The Doctor stated that the Cyberium had chosen her over the Cyberman, referring to it as Time Lord magnetism and remarked that it looked like she was its true guardian. After the Cyberman threatened to destroy Earth, the Doctor didn't want to take the risk and released the Cyberium to the Cyberman who took it back to the future, (TV: The Haunting of Villa Diodati) in the immediate aftermath of the Cyber-Wars.
As the Doctor reasoned, she was recognised by the Cyberium's memory as an enemy of the Cybermen. Believing the Cyberium to see all, the Lone Cybermen felt that it had given him understanding and distilled his purpose, leading him to regard himself as the perfect vessel for the ascension of the Cybermen and the death of their enemies. With the knowledge of the Cyberium, the Cyberman and his Cyberguards found an abandoned Cybercarrier containing hundreds of thousands of dormant warriors. (TV: Ascension of the Cybermen)
The Master was later interested to learn that Ashad was the host of the Cyberium which had created the death particle for him. After Ashad stated that the Cyberium would never leave him while Ashad was still alive, the Master killed the Lone Cyberman with his Tissue Compression Eliminator, releasing the Cyberium. The Master proposed an alliance between them, pointing out that it had all the knowledge of the Cybermen while he had all of the knowledge of the Time Lords taken from the Matrix. The Cyberium agreed and took the Master as his host which he later revealed to the Doctor. However, Ko Sharmus, who felt responsible as he was part of the force that sent the Cyberium back through time and space, sacrificed himself to stop the Master and his new army by detonating the death particle, presumably killing the Master and his army. It is unknown what became of the Cyberium after this. (TV: The Timeless Children)